Average of a function. Integral form of the Mean Value Theorem. Polar coordinates.

Transcription

1 Math 20B Integral Calculus Lecture 6 1 Miscellaneous topics Slide 1 Average of a function. Integral form of the Mean Value Theorem. Polar coordinates. Integration provides a way to define the average of a continuous function on a closed interval Slide 2 Definition 1 The average of a continuous function f(x) in [a, b] is f ave = 1 b f(x) dx. b a a Theorem 1 Let m and M be the minimum and maximum values of a continuous function f(x) in [a, b]. Then, m f ave M.

2 Math 20B Integral Calculus Lecture 7 2 The Mean Value Theorem in differential form Theorem 2 If f(x) is differentiable in [a, b] with f (x) continuous in [a, b], then there exists c (a, b) such that f(b) f(a) = f (c) (b a). Slide 3 The Mean Value Theorem in integral form Theorem 3 If f(x) is continuous in [a, b], then there exists c (a, b) such that b f(x) dx = f(c) (b a). a Polar coordinates are useful to describe situations with circular symmetry Slide 4 Definition 2 Let (x, y) be Cartesian coordinates in IR 2. Then, polar coordinates (r, θ) are given by ( y r = x 2 + y 2, θ = arctan. x) The inverse expression is x = r cos(θ), y = r sin(θ).

3 Math 20B Integral Calculus Lecture 7 3 Integration in polar coordinates Slide 5 Review of polar coordinates. Areas of angular sections. Riemann sums and integrals in polar coordinates. Examples. Review of polar coordinates Slide 6 Definition 3 Let (x, y) be Cartesian coordinates in IR 2. Then, polar coordinates (r, θ) are given by ( y r = x 2 + y 2, θ = arctan. x) The inverse expression is x = r cos(θ), y = r sin(θ).

4 Math 20B Integral Calculus Lecture 7 4 We review the formula for the area of circular sections The Greeks proved that for every disk or radius r, circumference, and area A T, the quotients Slide 7 2r, A T r 2 are equal and the same for every disk of any radius r. They called that constant π. Therefore, we have 2r = A T r 2 = π. Angles are defined as a fraction of the circumference length Slide 8 Let l be a portion of a circumference in a disk of radius r. Sexagesimal grads are defined as: Radians are defined as: ˆθ l = 360 l, ˆθl [0, 360]. θ l = l r, θ l [0, 2π]. ( θ l = l r = LT r l = 2π l ).

5 Math 20B Integral Calculus Lecture 7 5 The area of a circular section of a disk is proportional to the total area Slide 9 therefore, A θl = l r A θl = l A T, r A T = θ l 1 2π A T = θ l 1 2π πr2, so we finally get the area of a circular section to be: A θ = 1 2 r2 θ. Riemann sums in polar coordinates Slide 10 Let r, θ be polar coordinates in IR 2. Definition 4 Let r(θ) be a function defined on a interval θ [θ a, θ b ]. The integral of r(θ) in [θ a, θ b ] is the number given by 1 θb [r(θ)] 2 dθ = 1 n 1 2 θ a 2 lim [r(θ n i )]2 θ, if the limit exists. Given a natural number n we have introduced a partition on [θ a, θ b ] given by θ = (θ b θ a )/n. We denoted θ i = (θ i + θ i 1 )/2, where θ i = θ a + i θ, i = 0, 1,, n. i=0 This choice of the sample point θi is called midpoint rule.

6 Math 20B Integral Calculus Lecture 8 6 Comparison between integral in Cartesian and in polar coordinates The area below y = f(x) for x [a, b] is Slide 11 A = b a f(x) dx. The area below r = g(θ) for θ [θ a, θ b ] is A = 1 2 θb θ a [g(θ)] 2 dθ. Area between two functions in polar coordinates Slide 12 Given two functions r 2 (θ) r 1 (θ) the area in between these functions in the interval [θ a, θ b ] is A = 1 2 θb θ a ( [r2 (θ)] 2 [r 1 (θ)] 2) dθ.

7 Math 20B Integral Calculus Lecture 8 7 Magic numbers Slide 13 Introduction to complex numbers. Addition, multiplication, conjugate. Polar form, power of complex numbers. What if we create something? Slide 14 There is no real number solution of x 2 = 1. So the symbol 1 means nothing. What if we create a symbol for that nothingness, say, 2 = 1; we decide that satisfies every law of addition and multiplication that real numbers do satisfy. Is the result of that creation, meaningful?

8 Math 20B Integral Calculus Lecture 8 8 Complex numbers are really magic numbers It can be proved that our creation is free of internal contradictions. Slide 15 Several incredible beautiful theorems can be proved. These imaginary number are essential to describe the real world. Quantum mechanics cannot be created without complex numbers. Basic operations with complex numbers Let z = a + bi, and w = c + di be two complex numbers, a, b, c, d be real numbers. Then we define Slide 16 cz = ca + cbi; z + w = (a + c) + (b + d)i; zw = (ac bd) + (ad + bc)i; z = a bi; z 2 = zz = a 2 + b 2.

9 Math 20B Integral Calculus Lecture 8 9 Notice these strange properties of our creation Slide 17 i 0 = 1, i 1 = i, i 2 = 1, i 3 = i, i 4 = 1, i 5 = i. Also: i = 1 i. Complex numbers can be represented as points in a plane Slide 18 Cartesian picture: Good for representing addition and real umber multiplication. (Parallelogram law and stretching.) Polar picture: Good for representing the (weird) multiplication law. (Stretching and rotation.)